{
 "cells": [
  {
   "cell_type": "code",
   "id": "initial_id",
   "metadata": {
    "collapsed": true,
    "ExecuteTime": {
     "end_time": "2025-04-18T08:36:29.144784Z",
     "start_time": "2025-04-18T08:36:29.140640Z"
    }
   },
   "source": [
    "#演示一下过拟合欠拟合，用多项式\n",
    "import math\n",
    "import numpy as np\n",
    "import torch\n",
    "from torch import nn\n",
    "from d2l import torch as d2l\n",
    "from utils import Utils as ut"
   ],
   "outputs": [],
   "execution_count": 16
  },
  {
   "metadata": {
    "ExecuteTime": {
     "end_time": "2025-04-18T08:36:29.175961Z",
     "start_time": "2025-04-18T08:36:29.161605Z"
    }
   },
   "cell_type": "code",
   "source": [
    "max_degree = 20  # 多项式的最大阶数\n",
    "n_train, n_test = 100, 100  # 训练和测试数据集大小\n",
    "true_w = np.zeros(max_degree)  # 分配大量的空间\n",
    "true_w[0:4] = np.array([5, 1.2, -3.4, 5.6])\n",
    "\n",
    "features = np.random.normal(size=(n_train + n_test, 1))\n",
    "np.random.shuffle(features)\n",
    "poly_features = np.power(features, np.arange(max_degree).reshape(1, -1))\n",
    "for i in range(max_degree):\n",
    "    poly_features[:, i] /= math.gamma(i + 1)  # gamma(n)=(n-1)!\n",
    "# labels的维度:(n_train+n_test,)\n",
    "labels = np.dot(poly_features, true_w)\n",
    "labels += np.random.normal(scale=0.1, size=labels.shape)"
   ],
   "id": "dfc9c99c5d7ccaab",
   "outputs": [],
   "execution_count": 17
  },
  {
   "metadata": {
    "ExecuteTime": {
     "end_time": "2025-04-18T08:36:29.206459Z",
     "start_time": "2025-04-18T08:36:29.192420Z"
    }
   },
   "cell_type": "code",
   "source": [
    "# NumPy ndarray转换为tensor\n",
    "true_w, features, poly_features, labels = [torch.tensor(x, dtype=\n",
    "    torch.float32) for x in [true_w, features, poly_features, labels]]\n",
    "\n",
    "features[:2], poly_features[:2, :], labels[:2]"
   ],
   "id": "d22d94940f678ece",
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(tensor([[-0.9939],\n",
       "         [-1.2584]]),\n",
       " tensor([[ 1.0000e+00, -9.9393e-01,  4.9395e-01, -1.6365e-01,  4.0664e-02,\n",
       "          -8.0834e-03,  1.3391e-03, -1.9013e-04,  2.3622e-05, -2.6088e-06,\n",
       "           2.5929e-07, -2.3429e-08,  1.9406e-09, -1.4837e-10,  1.0533e-11,\n",
       "          -6.9796e-13,  4.3358e-14, -2.5350e-15,  1.3998e-16, -7.3225e-18],\n",
       "         [ 1.0000e+00, -1.2584e+00,  7.9178e-01, -3.3212e-01,  1.0448e-01,\n",
       "          -2.6297e-02,  5.5152e-03, -9.9148e-04,  1.5596e-04, -2.1806e-05,\n",
       "           2.7441e-06, -3.1392e-07,  3.2920e-08, -3.1866e-09,  2.8643e-10,\n",
       "          -2.4029e-11,  1.8899e-12, -1.3990e-13,  9.7802e-15, -6.4775e-16]]),\n",
       " tensor([ 1.1538, -1.0965]))"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "execution_count": 18
  },
  {
   "metadata": {
    "ExecuteTime": {
     "end_time": "2025-04-18T08:36:29.238312Z",
     "start_time": "2025-04-18T08:36:29.223361Z"
    }
   },
   "cell_type": "code",
   "source": [
    "def evaluate_loss(net, data_iter, loss):  #@save\n",
    "    \"\"\"评估给定数据集上模型的损失\"\"\"\n",
    "    metric = d2l.Accumulator(2)  # 损失的总和,样本数量\n",
    "    for X, y in data_iter:\n",
    "        out = net(X)\n",
    "        y = y.reshape(out.shape)\n",
    "        l = loss(out, y)\n",
    "        metric.add(l.sum(), l.numel())\n",
    "    return metric[0] / metric[1]"
   ],
   "id": "2323dee58a5c1a1a",
   "outputs": [],
   "execution_count": 19
  },
  {
   "metadata": {
    "ExecuteTime": {
     "end_time": "2025-04-18T08:36:29.269833Z",
     "start_time": "2025-04-18T08:36:29.254891Z"
    }
   },
   "cell_type": "code",
   "source": [
    "def train(train_features, test_features, train_labels, test_labels,\n",
    "          num_epochs=400):\n",
    "    loss = nn.MSELoss(reduction='none')\n",
    "    input_shape = train_features.shape[-1]\n",
    "    # 不设置偏置，因为我们已经在多项式中实现了它\n",
    "    net = nn.Sequential(nn.Linear(input_shape, 1, bias=False))\n",
    "    batch_size = min(10, train_labels.shape[0])\n",
    "    train_iter = d2l.load_array((train_features, train_labels.reshape(-1,1)),\n",
    "                                batch_size)\n",
    "    test_iter = d2l.load_array((test_features, test_labels.reshape(-1,1)),\n",
    "                               batch_size, is_train=False)\n",
    "    trainer = torch.optim.SGD(net.parameters(), lr=0.01)\n",
    "    animator = d2l.Animator(xlabel='epoch', ylabel='loss', yscale='log',\n",
    "                            xlim=[1, num_epochs], ylim=[1e-3, 1e2],\n",
    "                            legend=['train', 'test'])\n",
    "    for epoch in range(num_epochs):\n",
    "        ut.train_epoch_ch3(net, train_iter, loss, trainer)\n",
    "        if epoch == 0 or (epoch + 1) % 20 == 0:\n",
    "            animator.add(epoch + 1, (evaluate_loss(net, train_iter, loss),\n",
    "                                     evaluate_loss(net, test_iter, loss)))\n",
    "    print('weight:', net[0].weight.data.numpy())"
   ],
   "id": "f143b2ed79a0ccbe",
   "outputs": [],
   "execution_count": 20
  },
  {
   "metadata": {},
   "cell_type": "code",
   "source": [
    "# 从多项式特征中选择前4个维度，即1,x,x^2/2!,x^3/3!\n",
    "train(poly_features[:n_train, :4], poly_features[n_train:, :4],\n",
    "      labels[:n_train], labels[n_train:])"
   ],
   "id": "7cb27770386d2288",
   "execution_count": 21,
   "outputs": [
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  },
  {
   "metadata": {},
   "cell_type": "code",
   "source": [
    "# 从多项式特征中选择前2个维度，即1和x\n",
    "train(poly_features[:n_train, :2], poly_features[n_train:, :2],\n",
    "      labels[:n_train], labels[n_train:])"
   ],
   "id": "4b4f9f13a26a9f3a",
   "execution_count": 22,
   "outputs": [
    {
     "data": {
      "text/plain": [
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  },
  {
   "metadata": {},
   "cell_type": "code",
   "source": [
    "# 从多项式特征中选取所有维度\n",
    "train(poly_features[:n_train, :], poly_features[n_train:, :],\n",
    "      labels[:n_train], labels[n_train:], num_epochs=1500)"
   ],
   "id": "7a241d97d2606488",
   "execution_count": 23,
   "outputs": [
    {
     "data": {
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